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Math Help - Marginals

  1. #1
    Newbie
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    Marginals

    I need some help with the limits of the integrals when you work out the marginal pdf of X or Y.

    I've got the formulae

    f(x)= \int_{-\infty}^{\infty}f(x,y)dy
    for the marginal pdf of X and

    f(y)= \int_{-\infty}^{\infty}f(x,y)dx
    for Y

    Then there's the question
    Random variables X and Y have joint pdf

     <br />
f(x,y)=\left\{\begin{array}{cc}xy,&\mbox{for}<br />
(x,y)inA\\0, & \mbox{ elsewhere } \end{array}\right.<br />


    where A is the region bounded between y=x and y=x^2

    Looking at the solutions, the limits for f(x) are between x and x^2 and I think I get that one but for f(y) they're between y and y^\frac{1}{2} which I can't figure out. Can anyone help?
    Last edited by Nightfly; November 20th 2008 at 11:10 AM.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Nightfly View Post
    I need some help with the limits of the integrals when you work out the marginal pdf of X or Y.

    I've got the formulae

    f(x)= \int_{-\infty}^{\infty}f(x,y)dy
    for the marginal pdf of X and

    f(y)= \int_{-\infty}^{\infty}f(x,y)dx
    for Y

    Then there's the question
    Random variables X and Y have joint pdf

     <br />
f(x,y)=\left\{\begin{array}{cc}xy,&\mbox{for}<br />
(x,y)inA\\0, & \mbox{ elsewhere } \end{array}\right.<br />


    where A is the region bounded between y=x and y=x^2

    Looking at the solutions, the limits for f(x) are between x and x^2 and I think I get that one but for f(y) they're between y and y^\frac{1}{2} which I can't figure out. Can anyone help?
    The marginal pdf for y is:

    f(y)= \int_A f(x,y)dx= \int_{x=y}^{\sqrt{y}} xy \ dx=<br />
\left[ \frac{x^2y}{2}\right]_{x=y}^{\sqrt{y}},\ \ y \in (0,1)<br />

    CB
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